#239885

order	512
nilpotency class	3
isoclinism	isoclinic class of #233958
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35191	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,10)(8,16)(9,13)(11,15)(12,18)(14,19)(17,20)(21,22)(23,27)(24,28)(25,33)(26,29)(30,35)(31,36)(32,34);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,16)(12,20)(13,19)(17,18)(21,23)(22,27)(24,32)(25,30)(26,31)(28,34)(29,36)(33,35);(1,3,4,2)(5,7)(6,11)(8,17)(9,14)(10,16)(12,19)(13,20)(15,18)(23,30)(24,25)(27,34)(28,33)(31,32)(35,36);(7,14)(10,19)(11,20)(15,17)(21,24,26,25)(22,28,29,33)(23,30,31,32)(27,35,36,34)

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147081);